Theory of parametric x-ray optical wavemixing processes

Enabled by x-ray free-electron lasers, nonlinear optical phenomena can be explored in the x-ray domain nowadays. Among the multitude of newly accessible processes, this theoretical study focuses parametric x-ray optical wavemixing for closer investigation. Specifically, we develop a framework based on non-relativistic QED to describe x-ray optical sum- and difference-frequency generation as well as x-ray parametric down-conversion on equal footing. All of these processes promise imaging capabilities similar to regular x-ray diffraction with additional spectroscopic selectivity that is tunable via the optical admixture. Based on our derivation, we identify the imaged quantity as we relate the observable scattering pattern to an underlying response function of the medium. The resulting relation, furthermore, enables the microscopic reconstruction of this response function from nonlinear analogues of crystallographic measurements. We benchmark our approach on recent experimental sum-frequency results, for which we find encouraging agreement with our theory.